For any sequence \(\lbrace \pi _k\rbrace _{k\in \mathbb {N}}\in \widehat{\Pi }\)  that converges to the supremum value, that is
\(\lim _{k\rightarrow \infty }V^{\mu }(\pi _k)=\sup _{\pi \in \widehat{\Pi }}V^\mu (\pi )\) , there always exists a \(\gamma \in \Gamma \) , such that \(\mathcal {P}^\gamma \)  include a subsequence of \(\lbrace \pi _k\rbrace _{k\in \mathbb {N}}\) . This is because if every \(\mathcal {P}^\gamma \)  only comprises a finite part of the sequence, then the original sequence would be finite, as there are only finite partitions, contradicting the definition of supremum. Without loss of generality, we still denote the subsequence by \(\lbrace \pi _k\rbrace _{k\in \mathbb {N}}\) , and a \(\pi _K\)  with sufficiently large \(K\in \mathbb {N}\)  is naturally an \(\epsilon -\)  equilibrium.

It should be noted that in basic Bayesian Stackelberg game, the notion of approximate equilibrium is generally not used in the related literatureAs a solution concept per se, the approximate equilibrium may not carry significant meaning in Stackelberg games [1]}. However, we must point out that from the perspective of algorithmic game theory, the approximability results are important both in theory, and practice [2]}., as the communication between players and the information asymmetry is one-sided: the leader makes her commitment with her incomplete knowledge about the follower, and the follower is purely receiver. Hence, from the leader's perspective, when announcing the commitment, she must take into account the possible best responses of followers of various types, which is generally described by a set of linear constraints (see the programming-based computation approaches in [3]}, [4]}). However, in our proposed Stackelberg game model, the approximate equilibrium arises from the two-way communication and double-sided information asymmetry, where the leader (the follower) reveals partial information regarding her realized action (his private type), which is hidden from the other side. In the presence of the follower's misreporting, as argued in lem:linear and its subsequent discussion, the leader may not be able to achieve the supremum in the end, and the approximate equilibrium is the best she can hope.
